【作者】 刘芳；

【导师】 杨孝平；

【作者基本信息】 南京理工大学 ， 系统工程， 2013， 博士

【摘要】 无穷Laplace方程涉及变分法、泛函分析、微分几何以及拟线性偏微分方程等重要研究领域。该类方程的研究起源于L∞变分问题,在博弈论、形变、最优传输、图像处理、弹性力学及物理等方面有着广泛的应用。经典的无穷Laplace方程是一类拟线性、高度退化的偏微分方程,形如近二十年来,对于无穷Laplace方程的研究取得了丰硕的成果。本文主要涉及无穷Laplace方程的四个方面的研究：一是讨论一类含有无穷Laplace的非齐次椭圆方程的Dirichlet边值问题粘性解的存在性及唯一性,同时我们对解在孤立奇点附近的性质进行了研究,我们也对一类含无穷Laplace算子的椭圆方程的光滑解得到了先验估计；二是讨论一类含有无穷Laplace算子的齐次抛物方程的初边值问题,得到了正解的唯一性,利用超几何函数得到了方程的一类特殊形式的解,同时分析了解的渐近行为；三是讨论一类含有正规化无穷Laplace算子的非齐次抛物方程的初边值问题,我们利用标准的粘性解扰动理论得到了解的唯一性,利用正则化方程逼近的方法得到了解的存在性；四是讨论正规化p-Laplace的非齐次抛物方程,我们利用扰动理论得到了其初边值问题解的唯一性,同时也给出了方程的解在粘性意义下的渐近平均值公式。本文研究几类含有无穷Laplace算子的方程,所得的主要结论如下：(1)研究一类椭圆型非齐次无穷Laplace方程的Dirichlet:边值问题,用经典的Perron方法证明该问题粘性解的存在性,然后给出几类非齐次无穷Laplace方程的解在孤立奇点附近的渐近行为,最后对一类无穷Laplace方程的古典解给出先验估计。(2)讨论一类齐次抛物无穷Laplace方程的初边值问题正解的唯一性及解的渐近行为,并给出了方程的一类分离变量形式的解。(3)研究一类含有正规化无穷Laplace的非齐次抛物方程,证明了该方程的初边值问题粘性解的比较原理,得到了解的唯一性,同时用一致估计的方法得到了解的存在性结果。(4)研究含有正规化p-Laplace算子的非齐次抛物方程,给出了比较原理及解在粘性意义下的渐近平均值公式。本文所采用的主要方法是：Perron方法、闸函数法、一致估计法、分离变量法及扰动法。更多还原

【Abstract】 Infinity Laplacian equations are very important topics that deal with calculus of variations, functional analysis, differential geometry and quasi-linear partial differential equations etc. L∞minimal problem is the source of infinity Laplacian equations. The infinity Laplacian equa-tions have been widely applied in game theory, shape deformation, optimal transport, image processing, elastic mechanics, physics and other fields.The classical form of infinity Laplacian equations is a quasi-linear and highly degenerate partial differential equation:During the last twenty years infinity Laplacian equations have been a subject of intensive studies. This thesis mainly studies four aspects as follows:First, existence and uniqueness of viscosity solutions for Dirichlet problem of some elliptic inhomogeneous infinity Laplacian equation are discussed, and then the properties of solutions near an isolated singularity and the priori estimates of the smooth solutions for some elliptic infinity Laplacian equation are stud-ied; Second, the uniqueness of the positive solutions for the initial-boundary value problem of a homogeneous parabolic infinity Laplacian equation is discussed, a kind of solutions of special form is obtained by super-geometric functions and the asymptotic behavior is also discussed; Third, existence and uniqueness of viscosity solutions for the initial-boundary value problem of a parabolic inhomogeneous infinity Laplacian equation are discussed by the method of reg-ularized equations and standard perturbation theory of viscosity solutions respectively; Finally, the uniqueness of viscosity solutions for the initial-boundary value problem of a parabolic in-homogeneous normalized p-Laplace equation and the asymptotic mean value formula of the solutions in the viscosity sense are studied.Some classes of equations involving infinity Laplace operator are studied in this paper. The main results achieved include:(1) We prove the existence of viscosity solutions for Dirichlet problem of an elliptic inho-mogeneous infinity Laplacian equation by the classical Perron’s method. Then we prove some properties of the solutions near an isolated singularity. Finally, we obtain the priori estimates of the smooth solutions for some elliptic infinity Laplacian equation.(2) We prove the uniqueness of the positive solutions for the initial-boundary value prob-lem of a homogeneous parabolic infinity Laplacian equation. Then we give some special solu-tions by super-geometric functions. Finally we give the asymptotic behavior of the solutions.(3) We prove the comparison principle and uniqueness of viscosity solutions for the initial-boundary value problem of a parabolic inhomogeneous infinity Laplacian equation. Then we prove the existence of the solutions by the uniform estimates.(4) We prove the comparison principle of viscosity solutions for the initial-boundary value problem of a parabolic inhomogeneous normalized p-Laplace equation. Then we give the asymptotic mean value formula of the solutions in the viscosity sense.In this thesis, we mainly use the methods including Perron’s method, barrier functions, a uniform estimate, separation of variables and perturbation method.更多还原